3.593 \(\int (a+b x^n)^p (c+d x^n)^p (e+\frac{(b c+a d) e (1+n+n p) x^n}{a c}+\frac{b d e (1+2 n+2 n p) x^{2 n}}{a c}) \, dx\)

Optimal. Leaf size=31 \[ \frac{e x \left (a+b x^n\right )^{p+1} \left (c+d x^n\right )^{p+1}}{a c} \]

[Out]

(e*x*(a + b*x^n)^(1 + p)*(c + d*x^n)^(1 + p))/(a*c)

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Rubi [A]  time = 0.205318, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 69, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.014, Rules used = {1897} \[ \frac{e x \left (a+b x^n\right )^{p+1} \left (c+d x^n\right )^{p+1}}{a c} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^n)^p*(c + d*x^n)^p*(e + ((b*c + a*d)*e*(1 + n + n*p)*x^n)/(a*c) + (b*d*e*(1 + 2*n + 2*n*p)*x^(2*n
))/(a*c)),x]

[Out]

(e*x*(a + b*x^n)^(1 + p)*(c + d*x^n)^(1 + p))/(a*c)

Rule 1897

Int[((a_) + (b_.)*(x_)^(n_.))^(p_.)*((c_) + (d_.)*(x_)^(n_.))^(p_.)*((e_) + (f_.)*(x_)^(n_.) + (g_.)*(x_)^(n2_
.)), x_Symbol] :> Simp[(e*x*(a + b*x^n)^(p + 1)*(c + d*x^n)^(p + 1))/(a*c), x] /; FreeQ[{a, b, c, d, e, f, g,
n, p}, x] && EqQ[n2, 2*n] && EqQ[a*c*f - e*(b*c + a*d)*(n*(p + 1) + 1), 0] && EqQ[a*c*g - b*d*e*(2*n*(p + 1) +
1), 0]

Rubi steps

\begin{align*} \int \left (a+b x^n\right )^p \left (c+d x^n\right )^p \left (e+\frac{(b c+a d) e (1+n+n p) x^n}{a c}+\frac{b d e (1+2 n+2 n p) x^{2 n}}{a c}\right ) \, dx &=\frac{e x \left (a+b x^n\right )^{1+p} \left (c+d x^n\right )^{1+p}}{a c}\\ \end{align*}

Mathematica [A]  time = 0.523959, size = 31, normalized size = 1. \[ \frac{e x \left (a+b x^n\right )^{p+1} \left (c+d x^n\right )^{p+1}}{a c} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^n)^p*(c + d*x^n)^p*(e + ((b*c + a*d)*e*(1 + n + n*p)*x^n)/(a*c) + (b*d*e*(1 + 2*n + 2*n*p)*
x^(2*n))/(a*c)),x]

[Out]

(e*x*(a + b*x^n)^(1 + p)*(c + d*x^n)^(1 + p))/(a*c)

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Maple [A]  time = 0.153, size = 52, normalized size = 1.7 \begin{align*}{\frac{xe \left ( bd \left ({x}^{n} \right ) ^{2}+ad{x}^{n}+bc{x}^{n}+ac \right ) \left ( a+b{x}^{n} \right ) ^{p} \left ( c+d{x}^{n} \right ) ^{p}}{ac}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*x^n)^p*(c+d*x^n)^p*(e+(a*d+b*c)*e*(n*p+n+1)*x^n/a/c+b*d*e*(2*n*p+2*n+1)*x^(2*n)/a/c),x)

[Out]

(a+b*x^n)^p*(b*d*(x^n)^2+a*d*x^n+b*c*x^n+a*c)*e*x/a/c*(c+d*x^n)^p

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Maxima [A]  time = 1.25654, size = 80, normalized size = 2.58 \begin{align*} \frac{{\left (b d e x x^{2 \, n} + a c e x +{\left (b c e + a d e\right )} x x^{n}\right )} e^{\left (p \log \left (b x^{n} + a\right ) + p \log \left (d x^{n} + c\right )\right )}}{a c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^n)^p*(c+d*x^n)^p*(e+(a*d+b*c)*e*(n*p+n+1)*x^n/a/c+b*d*e*(2*n*p+2*n+1)*x^(2*n)/a/c),x, algorit
hm="maxima")

[Out]

(b*d*e*x*x^(2*n) + a*c*e*x + (b*c*e + a*d*e)*x*x^n)*e^(p*log(b*x^n + a) + p*log(d*x^n + c))/(a*c)

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Fricas [A]  time = 1.79826, size = 115, normalized size = 3.71 \begin{align*} \frac{{\left (b d e x x^{2 \, n} + a c e x +{\left (b c + a d\right )} e x x^{n}\right )}{\left (b x^{n} + a\right )}^{p}{\left (d x^{n} + c\right )}^{p}}{a c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^n)^p*(c+d*x^n)^p*(e+(a*d+b*c)*e*(n*p+n+1)*x^n/a/c+b*d*e*(2*n*p+2*n+1)*x^(2*n)/a/c),x, algorit
hm="fricas")

[Out]

(b*d*e*x*x^(2*n) + a*c*e*x + (b*c + a*d)*e*x*x^n)*(b*x^n + a)^p*(d*x^n + c)^p/(a*c)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x**n)**p*(c+d*x**n)**p*(e+(a*d+b*c)*e*(n*p+n+1)*x**n/a/c+b*d*e*(2*n*p+2*n+1)*x**(2*n)/a/c),x)

[Out]

Timed out

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Giac [B]  time = 1.17098, size = 155, normalized size = 5. \begin{align*} \frac{{\left (b x^{n} + a\right )}^{p}{\left (d x^{n} + c\right )}^{p} b d x x^{2 \, n} e +{\left (b x^{n} + a\right )}^{p}{\left (d x^{n} + c\right )}^{p} b c x x^{n} e +{\left (b x^{n} + a\right )}^{p}{\left (d x^{n} + c\right )}^{p} a d x x^{n} e +{\left (b x^{n} + a\right )}^{p}{\left (d x^{n} + c\right )}^{p} a c x e}{a c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^n)^p*(c+d*x^n)^p*(e+(a*d+b*c)*e*(n*p+n+1)*x^n/a/c+b*d*e*(2*n*p+2*n+1)*x^(2*n)/a/c),x, algorit
hm="giac")

[Out]

((b*x^n + a)^p*(d*x^n + c)^p*b*d*x*x^(2*n)*e + (b*x^n + a)^p*(d*x^n + c)^p*b*c*x*x^n*e + (b*x^n + a)^p*(d*x^n
+ c)^p*a*d*x*x^n*e + (b*x^n + a)^p*(d*x^n + c)^p*a*c*x*e)/(a*c)